Would casting the shield spell allow you to get bonus saving throw to spells targeting you or to an effect that would damage you?
While the name of the spell is shield, it does not create one, but
An invisible barrier of magical force (PHB 275)
The description is always the place where you can find how the spell fits in the rules, mechanical effects are always mentioned there, directly or indirectly. Interacting with other features is a mechanical effect.
Shield specifically adds to your Armor Class and not your Saving Throw bonuses:
Until the start of your next turn, you have a +5 bonus to AC
Spells do what they say they do on the tin, no more no less, and that means, though a bit weird, Shield will only protect you from attacks. If it is a harmful effect that requires a Save (like a Fireball or Disintigrate), Shield is no use.
The Shield spell, despite being a barrier, is not an actual "shield" in terms of what the game mechanics recognize as such, so no. Shield Master also is specific with its wording:
You gain the following benefits while you are wielding a shield:
The use of the words "wielding" and "a" indicate quite specifically that it must be a shield armament and cannot include the Shield spell.